Analysis is conducted on the non-trivial different integral solution to the quadratic equation . We derive distinct integral solutions in four different patterns. There are a few intriguing connections between the solutions and unique polygonal numbers that are presented.
The ternary quadratic equation has four different patterns of non- zero distinct integral solutions, which we described in this paper. For other quadratic equation , one can look for other patterns of non-zero integer unique solutions and their accompanying features.
 Dickson L.E., “History of the theory of numbers”, Chelsia Publishing Co., Vol II, New York, 1952.
 Carmichael R.D., “The Theory of Numbers and Diophantine Analysis”, Dove Publications, New York, 1959.
 Mordell L .J, “Diophantine Equations”, Academic Press, London 1969.
 Telang S. G., “Number Theory”, Tata Mc Graw-Hill Publishing Company, New Delhi , 1996.
 Janaki G, Saranya C, “Integral Solutions of Binary Quadratic Diophantine Equation ”, International Journal of Scientific Research in Mathematical and Statistical Sciences , Vol 7, Issue II, Pg.No152-155, April 2020.
 Janaki G, Saranya C, “Observations on Ternary Quadratic Diophantine Equation ”, International Journal of Innovative Research and science, Engineering and Technology, Volume 5, Issue 2, February 2016.
 Janaki G, Saranya C, “ On the Ternary Quadratic Diophantine Equation ”, Imperical Journal of Interdisplinary Research , Vol 2, Feb 2016.
 Janaki G and Vidyalakshmi S, “Integral solutions of xy+x+y+1 = z2 - w2” , Antartica J.math , 7(1), 31-37, (2010).
 M.A. Gopalan and S.Vidyalakshmi, “Quadratic Diophantine equation with four variables x2+y2+xy+y=u2+v2uv+u-v”. Impact J. Sci. Tech: Vol 2(3), 125127, 2008
 Janaki G, Radha R, “On Ternary Quadratic Diophantine Equation ”, International Journal for Research in Applied Science and Engineering Technology, Vol 6, Issue I, January 2018.